GALOIS THEORY AND PAINLEVE EQUATIONS´ by Hiroshi Umemura · GALOIS THEORY AND PAINLEVE EQUATIONS´...

Seminaires & Congres

14, 2006, p. 299–339

GALOIS THEORY AND PAINLEVE EQUATIONS

by

Hiroshi Umemura

Abstract. — The paper consists of two parts. In the first part, we explain an excellent

idea, due to mathematicians of the 19-th century, of naturally developing classical

Galois theory of algebraic equations to an infinite dimensional Galois theory of non-

linear differential equations. We show with an instructive example how we can realize

the idea of the 19-th century in a rigorous framework. In the second part, we ask

questions arising from general Galois theory and Galois theoretic study of Painleve

equations. We also propose an infinite dimensional Galois theory of difference equa-

tions.

Résumé(Théorie de Galois et Équations de Painlevé). —Dans une premiere partie, nous

rappelons une excellente idee de mathematiciens du 19eme siecle en vue d’etendre la

theorie de Galois classique pour les equations algebriques en une theorie de Galois de

dimension infinie pour les equations differentielles non-lineaires. Nous illustrons par

un exemple instructif comment concretiser cette idee de facon rigoureuse.

Dans une deuxieme partie, nous formulons des questions liees a la theorie de

Galois generale et aux aspects galoisiens des equations de Painleve. Nous esquissons,

en outre, une theorie de Galois de dimension infinie pour les equations aux differences.

1. Introduction

Since Lie tried to apply the rich idea of Galois and Abel in algebraic equations to

analysis, Galois theory of differential equations has been attracting mathematicians.

Finite dimensional differential Galois theory was developed by Picard, Vessiot and

Kolchin and is widely accepted. As Lie already noticed it, the most important part

of differential Galois theory is, however, infinite dimensional. After a few trails have

been done about 100 years ago, the subject was almost forgotten. We proposed a

differential Galois theory of infinite dimension [14] in 1996 which is a Galois theory

2000Mathematics Subject Classification. — 34M55, 12H05, 12H10,13B05, 17B65,34M15, 58H05 .

Key words and phrases. — Galois theory, Painleve equations, Special functions, Differential algebra.

c© Seminaires et Congres 14, SMF 2006

300 H. UMEMURA

of differential field extension. On the other hand, a Galois theory of foliation by

B. Malgrange [11] that is also infinite dimensional, appeared in 2001. We do not feel

that they are well understood.

Our aim in Part I, Invitation to Galois theory, is to explain with examples that

our theory is a consequence of natural development of Galois theory of algebraic

equations. We recall how mathematicians of the 19-th century understood Galois

theory of algebraic equations and extend it to linear ordinary differential equations in

§§2 and 3. §4 is the most substantial section of the first part. We show a marvelous

idea of mathematicians of the 19-th century in Subsection 4.1 and realize it in the

framework of algebraic geometry. Since the reader can find rigorous reasonings in

[14], we repeatedly use a concrete and yet sufficiently general case, Instructive Case

(IC) in Subsection 4.4, to illustrate clearly what is going on.

In Part II, we ask questions about (1) general Galois theory and (2) Galois theoretic

study of Painleve equations. Among the questions about general Galois theory, we

cite descent of the field of definition of our Galois group Infgal(L/K) (Questions 1, 2

and 3) and comparison of Malgrange’s theory and ours (Question 4), while calculation

of Galois group of Painleve equations (Question 6), understanding of a remarkable

paper of Drach on the sixth Painleve equation (Questions 7, 8, ..., 11) and arithmetic

property of the sixth Paineve equation (Questions 17 and 18) belong to the questions

about Galois theoretic study of Painleve equations. We also propose a Galois theory

of difference equation of infinite dimension and calculation of Galois group for qP6

of Jimbo and Sakai (Question 12). We added a star to those questions that seem to

require a new idea. The mark is, however, nothing more than a personal impression

of the author.

PART I

INVITATION TO GALOIS THEORY

2. Galois theory of algebraic equations

The aim of the first part is to explain how an intuitive idea of Galois theory of

algebraic equations develops to infinite dimensional differential Galois theory of non-

linear differential equations. We described the latter rigorously in a general framework

[14]. In this note we try to be more intuitive than formal so that the reader can realize

how natural the basic idea of our theory is.

Principal homogeneous space is one of the main ingredients of Galois theory. Let

us start by recalling the definition.

SEMINAIRES & CONGRES 14

GALOIS THEORY AND PAINLEVE EQUATIONS 301

Definition 2.1. — Let G be a group operating on a set S. Then we say that the oper-

ation (G,S) is a principal homogeneous space if for an element s ∈ S, the map

G −→ S, g 7−→ gs

is bijective.

Inspired by Galois theory for algebraic equations, S. Lie had a plan to apply the rich

idea of Galois and Abel to differential equations. Galois theory of algebraic equations

is an ideal theory and it has been the model of generalizations. Let us go back to the

19-th century and see how the mathematicians of that time understood Galois theory

and how they tried to generalize it.

Let K be a field and let

(1) F (x) := a0xn + a1x

n−1 + · · · + an = 0, ai ∈ K, for 0 ≤ i ≤ n

a0 6= 0, be an algebraic equation with coefficients in K. We suppose for simplicity

the field K is of characteristic 0. We assume that the roots of the algebraic equation

(1) are distinct. Then the symmetric group Sn of degree n on the n letters

{1, 2, · · · , n}

operates on the set

S := {(x1, x2, · · · , xn) |F (xi) = 0, for 1 ≤ i ≤ n, xi 6= xj if i 6= j}

of ordered sets (x1, x2, · · ·xn) of roots as permutations of the roots and

(Sn, S)

is a principal homogeneous space.

The basic symmetric functions are expressed by coefficients.

n∑

i=1

xi = −a1

a0,

∑

1≤i<j≤n

xixj =a2

a0,

· · ·

x1x2 · · ·xn = (−1)n an

a0.

If there is no constraints among the roots

x1, x2, · · · , xn

with coefficients in K other than those that are a consequence of the relations above,

then the Galois group of equation (1) is the full symmetric group Sn. If there are

constraints, they determine a subgroup G of Sn, consisting of those elements leaving

SOCIETE MATHEMATIQUE DE FRANCE 2006

302 H. UMEMURA

all the constraints invariant, as Galois group of the algebraic equation (1). To be

more precise, let us consider all rational functions

Aα(X1, X2, · · · , Xn) ∈ K(X1, X2, · · · , Xn)

of variables X1, X2, · · · , Xn with coefficients in K indexed by an appropriate set I

such that

Aα(x1, x2, · · · , xn) ∈ K,

The constraints Aα(x) determine the Galois group G as a subgroup of the symmetric

group Sn consisting of elements of Sn leaving all the constraints Aα(x) invariant.

Namely

G := {g ∈ Sn | Aα(xg(1), xg(2), · · · , xg(n)) = Aα(x1, x2, · · · , xn) for all α ∈ I}

Let us illustrate this by an example. Let us consider the following algebraic equa-

tion over Q.

(2) x3 − 7x+ 7 = (x− x1)(x− x2)(x− x3) = 0.

Upon setting

x = (x1, x2, x3),

we have a constraint

D(x) := (x1 − x2)(x1 − x3)(x2 − x3) = ±7 ∈ Q.

D(x) takes value +7 or −7 according as the order of the roots. In fact, D(x)2 is, by

definition, the discriminant of the cubic equation (2) so that

D(x)2 = −4 × (−7)3 − 27 × 72 = 49.

Indeed, the discriminant of a cubic equation

x3 + ax+ b = 0

is equal to

−4a3 − 27b2.

The Galois group must leave the constraint

D(x) = (x1 − x2)(x1 − x3)(x2 − x3)

invariant so that the Galois group is a subgroup of the alternating group A3 ⊂ S3.

We can moreover show that the Galois group coincides with the alternating group A3.

We see how principal homogeneous spaces appear in this context. To this end, let

us set

S := {(x1, x2, x3) |F (xi) = 0},

S+ := {x ∈ S |D(x) = 7},

S− := {x ∈ S |D(x) = −7}

so that we have

S = S+ q S−.



The alternating group A3 operates on both sets S+, S− and

(A3, S+), (A3, S−)

are principal homogeneous spaces. We started from the principal homogeneous space

(S3, S)

and we decompose it to get two principal homogeneous spaces

(A3, S+), (A3, S−).

What makes Galois theory of algebraic equation useful is the fact that we have the

Galois correspondence. Let us come back to the algebraic equation (1). We denote

by K an algebraic closure of K. Let L be a subfield of K generated over K by all the

roots xi’s for 1 ≤ i ≤ n of the algebraic equation (1). Namely

L := K(x1, x2, · · · , xn) ⊂ K.

This type of field extension, a field extension generated over a field K by all the roots

of an algebraic equation with coefficients in K, is called a Galois extension. Let us

denote the Galois group of the equation (1) by G(L/K). We can show that the group

G (L/K) is isomorphic to the group Aut(L/K) of K-automorphisms of the field L so

that the group G depends only on the field extension L/K that the algebraic equation

(1) determines! We owe this eminent idea to Dedekind. Let M be an intermediate

field of the field extension L/K. Then since the coefficients of the algebraic equation

(1) are in K and hence in M and since

L = K(x1, x2, · · · , xn) = M(x1, x2, · · · , xn),

the field extension L/M is also Galois. Hence we can speak of the Galois group

G (L/M) of the field extension L/M , which is a subgroup of the Galois groupG (L/K).

We have thus defined a map ϕ from the set

Field (L/K)

of intermediate fields of the field extension L/K to the set of subgroups

Group (G)

of the Galois group G = G (L/K) sending an intermediate subfield M to the subgroup

G (L/M):

ϕ : Field (L/K) → Group (G).

Conversely letH be a subgroup of the Galois groupG = G (L/K). ThenH determines

an intermediate field

LH := {z ∈ L | g(z) = z for every element g ∈ H ⊂ G = Aut(L/K) }

consisting of those elements of the field L that are left invariant by all the element of

H that is a subgroup of the field automorphism group Aut(L/K).


304 H. UMEMURA

Theorem 2.2. — The mappings

ϕ : Field(L/K) → Group (G), ψ : Group (G) → Field (L/K)

give a 1:1 correspondence between the elements of the two sets

Field (L/K), Group (G).

Namely

ϕ ◦ ψ = id, ψ ◦ φ = id.

For an intermediate field M , the following two conditions are equivalent.

1. The extension M/K is Galois.

2. The corresponding subgroup N := G (L/M) is a normal subgroup of the Galois

group G = G (L/K).

When these equivalent conditions are satisfied, we have a natural group isomorphism

G/N ' G (M/K).

3. Picard-Vessiot Theory

An ordinary differential field (F, d) consists of a field F and a derivation d : F → F

on F . For an element a ∈ F we denote often d(a) by a′ and we use the following

notation a(2) = d(d(a)), a(3) = d(a(2)), · · · . An element a ∈ F is said to be a constant

if d(a) = 0. The set CF of constants of F forms a subfield of F called the field of

constants of F . When there is no danger of confusion, we do not make the derivation

d explicit. Now let K be an ordinary differential field of characteristic 0 and K a

differential overfield such that the field of constants CK coincides with CK . Given a

matrix A ∈Mn(K), we consider a system of linear differential equations

(3) Y ′ = AY,

where Y ∈ GLn(K). We denote by S the set of all the solutions of (3) in K. Namely,

S := {Y ∈ GLn(K) |Y ′ = AY }.

Lemma 3.1. — The following assertion holds.

1. For Y ∈ S, g ∈ GLn(CK), Y g ∈ S.

2. If Y1, Y2 ∈ S, then

Y1Y−12 ∈ GLn(CK)

.

Proof. — The first assertion is trivial. To prove the second, it is sufficient to notice

(Y −11 Y2)

′ = (Y −11 )′Y2 + Y −1

1 Y ′2 = (−Y −1

1 Y ′1Y

−11 )Y2 + Y −1

1 AY2

= (−Y −11 (AY1)Y

−11 )Y2 + Y −1

1 AY2 = 0.



Lemma (3.1) shows that the general linear group GLn(CK) = GLn(CK) operates

on the set S by right multiplication in such a manner that (GLn(CK), S) is a principal

homogeneous space as in the case of algebraic equations.

From now on, we take a solution Y ∈ S once for all. The choice of a solution does

not affect the argument below. Indeed, the other solutions are expressed as Y g for

g ∈ GLn(CK). As in the case of algebraic equations, if there is no constraints among

the entries of Y except for trivial constrains given by elements of K, then the Galois

group of the linear differential equation (3) is the full general linear group GLn(CK).

Otherwise, constraints determine a closed subgroup of the algebraic group GLn(CK)

consisting those elements of the algebraic group GLn(CK) leaving all the constraints

invariant as the Galois group of the linear differential equation (3). Here, we should

make the definition of the constraints clear. A constraint should be a rational function

A(Y, Y ′, Y (2), · · · )

with coefficients in K of the entries yij ’s and their derivatives for 1 ≤ i, j ≤ n such

that the value

A(Y, Y ′, Y (2), · · · ) ∈ K.

But, thanks to the differential equation (3), we can eliminate the derivatives

Y ′, Y (2), · · · . So a constraint is a rational function A(Y ) of the entries yij ’s for

1 ≤ i, j ≤ n of Y with coefficients in K such that A(Y ) ∈ K. In the most gen-

eral case there is no non-trivial constraint. Indeed, in that case, the entries yij

(1 ≤ i, j ≤ n) are algebraically independent over the base field K.

Let us consider, for example, the Bessel equation.

(4) y′′ + x−1y′ + (1 − ν2x−2)y = 0,

ν ∈ C being a complex parameter. We assume ν /∈ 1/2 + Z. If we write the Bessel

equation (4) in matrix form,[y11 y12y12 y22

]′

=

[0 1

−1 + ν2x−2 −x−1

] [y11 y12y21 y22

].

We have to clarify the differential fields. The base field K is the field C(x) of rational

functions with coefficients in C and the overfield K is the field of all the meromorphic

functions on a small neighborhood of 1 in the complex plane. For example an open disc

centered at 1 with radius 1 on the complex plane. The derivation d is the derivation

d/dx with respect to the independent variable x.

Lemma 3.2. — There exists a constant c ∈ CK such that

det Y = cx−1.

Proof. — We know

(det Y )−1(detY )′ = tr A = −x−1.


306 H. UMEMURA

So detY satisfies a linear homogeneous differential equation

(5) y′ + x−1y = 0

as well as x−1. Hence

(det Y )x ∈ L

is a constant so that

(det Y )x = c ∈ CL = CK .

Now det Y = cx−1 ∈ K gives a constraint. The Galois group G is a subgroup of

GL2(CK) consisting of those elements leaving detY invariant. Namely

G ⊂ {g ∈ GL2(CK) | det(Y g) = detY } = SL2(CK).

We can show that indeed we have

G = SL2(CK).

See Kolchin [10], Appendix.

Now we look at principal homogeneous spaces appearing here. Let us set

Sc = {Y ∈ S | det Y = cx−1}.

for c ∈ CK . Then the Galois group G = SL2(CK) operates on the set Sc by right

multiplication such that the operation

(SL2(CK), Sc)

is a principal homogeneous space for every c ∈ CK and

S =∐

c∈CK

Sc.

This is the coset decomposition GL2(CK)/SL2(CK). We started from the principal

homogeneous space

(GL2(K), S)

and arrived at the smaller principal homogeneous spaces

(SL2(CK), Sc)

just as in the case of algebraic equations.

Let us come back to the general linear differential equation (3) defined over the

differential field K. The field L := K(Y ) = K(yij)1≤i,j≤n is closed under the deriva-

tion so that K(Y )/K is a differential field extension, which is called a Picard-Vessiot

extension. We can show that the Galois group G is isomorphic to the automorphism

group Aut(L/K) of differential field extension. So the Galois group depends only

on the differential field extension L/K and we set G = G (L/K). In Picard-Vessiot

theory we have Galois correspondence.



Theorem 3.3. — Let L/K be a Picard-Vessiot extension with Galois group G. If the

field CK of constants of the base field K is algebraic closed, then the mappings in

theorem (2.2) give a 1:1 correspondence between the elements of two sets.

1. The set Field (L/K) of differential intermediate fields of the Picard-Vessiot ex-

tension L/K.

2. The set of closed algebraic subgroups of the Galois group G defined over CK .

For an differential intermediate field M , the following two conditions are equivalent.

1. The extension M/K is Picard-Vessiot.

2. The corresponding algebraic subgroup N = G(M/K) of G(L/K) is normal.

When these equivalent conditions are satisfied, then we have a natural isomorphism

G/N ' G (M/K).

Remark 3.4. — As the form of the linear differential equation shows, Picard-Vessiot

theory is a Galois theory on the general linear groupGLn(C) and its closed subgroups.

Such algebraic groups are called linear algebraic groups. Hence we can say that Picard-

Vessiot theory is a Galois theory on a linear algebraic group. We can construct a

similar theory on algebraic groups in general other that the linear algebraic groups.

This generalization was already known in the 19-th century and later worked out

thoroughly by Kolchin.

4. Non-linear differential equation

4.1. Idea of mathematicians of the 19-th century. — Let (K, d) be an ordi-

nary differential field of meromorphic functions over a complex domain so that the

derivation d of the differential field K is the derivation d/dx with respect to the in-

dependent variable x. One of the simplest examples is the field K = (C(x), d/dx) of

rational functions.

We want to define Galois group of a non-linear algebraic differential equation with

coefficients in K. To simplify the situation, we assume that the given algebraic dif-

ferential equation is solved by y(n). Namely,

(6) y(n) = A(x, y, y′, · · · , y(n−1)),

where A ∈ K(y, y′, · · · , y(n−1)) is a rational function of

y, y′, · · · , y(n−1)

with coefficients in K.

Definition 4.1. — A meromorphic function

F (X, Y, Y ′, · · · , Y (n−1))


308 H. UMEMURA

of (n + 1)-variables is a first integral of the algebraic differential equation (6) if for

every solution y(x) of the algebraic differential equation (6),

F (x, y(x), y′(x), · · · , y(n−1)(x))

is a constant, i.e., independent of x.

The following proposition is well-known.

Proposition 4.2. — The following conditions are equivalent.

1. F is a first integral of the algebraic differential equation (6).

2. F satisfies a linear partial differential equation

(7) LF = 0,

where

L := ∂/∂X + Y ′∂/∂Y + · · · + Y (n−1)∂/∂Y (n−2)

+A(X,Y, Y ′, · · · , Y (n−1))∂/∂Y (n−1).

Let

F = (F1, F2, · · · , Fn)

be an ordered set of n-independent first integrals of (6). Namely

F1, F2, · · · , Fn

be n-first integrals meromorphic over a sub-domain of Cn+1 such that the Jacobian

(8)J(F1, F2, · · · , Fn)

J(Y (0), Y (1), · · · , Y (n−1))= det[∂Fi/∂Y

(j)]1≤i≤n, 0≤j≤n−1 6= 0.

Given a set of constants c = (c0, c1, · · · , cn−1) ∈ Cn, we can solve by implicit function

theorem yi(x, c) for 0 ≤ i ≤ n− 1 such that

Fi(x, y0(x, c), · · · , yn−1(x, c)) = ci−1

for 1 ≤ i ≤ n. The assumption (8) that the Jacobian 6= 0 implies that we have

∂jy0(x, c)/∂xj = yj(x, c)

and

∂ny0/∂xn = A(x, y0, y

′0, · · · , y

(n−1)0 )

so that y0(c, x) is a solution of the algebraic differential equation (6).

Theorem 4.3. — Modulo implicit function theorem, which is transcendental by nature,

the following procedures are equivalent.

1. To find a general solution y(x, c) of the algebraic differential equation (6).

2. To find independent solutions Fi of linear partial equation LFi = 0 for 1 ≤ i ≤ n.



Proof. — We have shown above how the second condition implies the first condition.

Conversely, given a general solution y(x, c) of (6) so that the Jacobian

(9)J(y(x, c), y′(x, c), · · · , y(n−1)(x, c))

J(c1, c2, · · · , cn)6= 0.

We can express the constants ci’s as a function of the independent variable x and

y(x, c), y′(x, c), · · · , y(n−1)(x, c) by implicit function theorem so that the constants

ci’s are independent first integrals of (6) for 1 ≤ i ≤ n.

4.2. Advantage of passing from non-linear ordinary to linear partial

What is the advantage of passing from the non-linear ordinary differential equation

(6) to the partial linear differential equation (7)? We explain that the partial linear

equation (7) reveals the hidden symmetry of the algebraic differential equation (6)

that is indispensable for construction of a Galois theory. To this end we begin with

the following trivial fact. Let

G1, G2, · · · , Gm

be first integrals of (6) holomorphic on a domain W of C(n+1) and ϕ be a holomorphic

function on a domain U of Cm containing

G1(W ) ×G2(W ) × · · · ×Gm(W ) ⊂ Cm

so that we can compose functions to get

(10) ϕ(G1, Gi, · · · , Gm)

which is a holomorphic function on the domain W ⊂ Cn+1. Then it follows from

the definition that the composite holomorphic function (10) is a first integral. Let us

apply this to a set of n-independent first integrals and coordinate transformation of

n-variables. Let F := (F1, F2, · · · , Fn) be independent first integrals such that every

Fi is holomorphic on a domain W of Cn+1 for 1 ≤ i ≤ n. and let

u := (u1, u2, · · · , un) 7→ Φ(u) = (ϕ1(u), ϕ2(u), · · · , ϕn(u))

be a coordinate transformation of n-variables. To be more precise

Φ : U → V

is a biholomorphic isomorphism of two non-empty open subsets U, V of Cn. If

F(W ) ⊂ U ⊂ Cn,

then we can consider the composite function ϕi(F), which are holomorphic on W , for

every 1 ≤ i ≤ n so that

Φ(F) = (ϕ1(F), ϕ2(F), · · · , ϕn(F))

is an ordered set of independent first integrals holomorphic on W .

Let us set

S := {(F1, F2, · · · , Fn)|The F ′i s are independent first integrals for 1 ≤ i ≤ n }


310 H. UMEMURA

and denote the set of all the coordinate transformations of n-variables by Γn. Let

us try not to be too nervous about the domains of definition because our aim is to

understand the idea of mathematicians of the 19-th century. With respect to the

composition of two coordinate transformations, Γn is not a group but almost a group.

Indeed, we can not necessarily compose two local isomorphisms unless the second

transformation is regular on the image of the first transformation. Γn is an example

of Lie pseudo-groups. A similar problem arises if we say that the Lie pseudo-group

Γn operates on S. Anyhow Γn is almost a group, a Lie pseudo-group and it almost

operates, pseudo-operates, on S such that

(Γn, S)

is almost a principal homogeneous space. Now we are very close to the situations in

the Galois theory of algebraic equations and Picard-Vessiot theory. We replace Lie

pseudo-group by formal group for our personal taste (cf. Subsections 4.6, 4.7, ..., 4.9).

Let us choose a set of n-independent first integrals

F = (F1, F2, · · · , Fn).

If there are constraints among the partial derivatives

∂mFi/∂Xa∂Y b, a ∈ N, b ∈ Nn, m = a+ |b|, 1 ≤ i ≤ n,

they would determine Galois group of the algebraic differential equation (6) as a

Lie pseudo-subgroup of the Lie pseudo-group Γn. This is a beautiful idea of the

mathematicians of the 19-th century!

4.3. Criticism to the idea. — The idea explained above is remarkable. Yet there

are problems if we examine it closely.

1. After R. Dedekind, the Galois group is not attached to an algebraic equation

but to the field extension that the algebraic equation determines. We start

from a base field K for the non-linear ordinary differential equation (6). In the

transition from ordinary to partial, choice of the partial base field M is not

clear.

2. Even if we can properly choose the partial base field M , then the partial differ-

ential field extension

M < F > /M

depends on the choice of a set of n-independent first integrals

F = (F1, F2, · · · , Fn).

Here we denote by M < F > the partial differential field with derivations

{∂

∂X,∂

∂Y, · · · ,

∂

∂Y (n−1), }

generated by F1, F2, · · · , Fn over M .



Namely let F′ be another set of n-independent first integrals. The general

picture is that they are related through a coordinate transformation. There

exists a coordinate transformation Φ of n-variables such that

F′ = Φ(F).

Since the transformation Φ is transcendental or it involves power series, the field

extension M < F > /M is totally different from

M < Φ(F) >= M < F′ > /M.

Remark 4.4. — In one of the last versions of his book, Dirichlet’s Vorlesungen uber

Zahlentheorie, R. Dedekind arrived at the distinguished idea of attaching the Galois

group to the field extension that a given algebraic equation defines. Galois theory

is rich and has many aspects so that there are other interpretations than working

in the framework of field extensions. Our view point in [14] is a Galois theory of

differential field extensions, whereas B. Malgrange [11] proposes a Galois theory of

foliations.

4.4. How do we overcome these difficulties?— We were inspired by an idea of

Vessiot in one of his last articles published in 1946. We considered algebraic differential

equation (6). This means we are working with a differential equations over an algebraic

variety. The space of initial conditions at x = x0 is an algebraic variety X0 and the

differential equation describes a movement over an algebraic variety so that algebraic

rational functions on the space X0 of initial conditions are considered as natural first

integrals. Let K be a base field. Let us treat the convergent case where we assume,

as we did previously, that K is a differential field of meromorphic functions over a

complex domain U ⊂ C so that CK = C. We work in a slightly more general situation

than the algebraic differential equation (6). The most general setting is the following.

Let L be a differential field extension of the base field K. We assume that L is of

finite type over K as an abstract field extension so that

L = K(z1, z2, · · · , zm).

Hence, we have

(11) z′i = Fi(z1, z2, · · · , zm),

with

Fi(z1, z2, · · · , zm) ∈ K(z1, z2, · · · , zm)

for 1 ≤ i ≤ m. By localization, we may assume that

Fi(z1, z2, · · · , zm) ∈ K[z1, z2, · · · , zm

for 1 ≤ i ≤ m. Now, we consider a general solution

zi(c1, c2, · · · , cm : x) for 1 ≤ i ≤ m


312 H. UMEMURA

of equation (11) depending on the parameters ci associated with the initial conditions

zi(c : x0) = ci

at a general point x0 fixed once for all. In particular, we have an isomorphism of

differential fields

L = K(z1, z2, · · · , zm)

' K(z1(c : x), z2(c : x), · · · , zm(c : x)).

We have to be careful. Since the field extension L/K was first given, the generators

z1, z2, · · · , zm of L over K are not always algebraically independent over K. Hence,

we can not choose the constants ci’s arbitrarily for 1 ≤ i ≤ n. We illustrate the

idea mainly in the following particular case. Indeed, what is essential is involved in

this particular case and understanding this particular case allows us to write down a

general theory in the language of algebraic geometry. See Example 3 below.

Instructive Case (IC). — We assume that the following conditions are satisfied.

1. K = C(x).

2. L = K(z1, z2, · · · , zn) and the zi’s are algebraically independent over K for

1 ≤ i ≤ n.

3. z′i = Fi(z1, z2, · · · , zn) with Fi(z1, z2, · · · , zn) ∈ C[x, z1, z2, · · · , zn] for 1 ≤

i ≤ n.

Under these assumptions, the system of ordinary differential equation in condi-

tion 3 of (IC) describes a dynamical system on the affine space An. We notice that

the algebraic differential equation (6) is a particular instance satisfying these condi-

tions if A is a polynomial in C[x, y, y′, · · · , y(n−1)]. Now, we consider the partial

derivatives

∂mzi(c : x)/∂xj∂cI , j ∈ N, I ∈ Nn, m = j + |I|, 1 ≤ i ≤ n

with respect to the independent variable x and the initial conditions c1, c2, · · · , cn.

Since we can eliminate the derivation ∂/∂x by virtue of the differential equation (11),

we have to consider only the derivatives

(12) ∂|I|zi(c, x)/∂cI , I ∈ Nn, 1 ≤ i ≤ n

with respect to the initial condition c. If there is no algebraic relations or if there is no

constraints among the derivatives (12) with coefficients in the field K(c) of rational

functions, then the Galois group of the differential field extension

L/K



is the full Lie pseudo-group Γn of all the coordinate transformations of the space An

of initial conditions. We are soon going to replace the Lie pseudo-group by an auto-

morphism group. So let us set

(13) G-Gal(L/K) := {Transformations c 7→ Φ(c)

leaving all the constraints invariant}.

Now, we can clarify the transition from non-linear ordinary to partial linear in terms

of differential field extension. We start from the ordinary differential field extension

L/K with derivation d = d/dx and arrived at the partial differential field extension

(14) K(c, ∂|I|zi/∂cI)I∈Nn/K(c)

with derivations

d = ∂/∂x, and ∂/∂ci, 1 ≤ i ≤ n.

We shall see later that we have to replace more correctly the partial differential field

extension (14) by a partial differential algebra extension (cf. Remark 4.8, (1)).

4.5. Examples

Example 1.— Let us take the simplest example of linear ordinary equations

(15) z′ = z

over the base field K = C(x) with derivation d/dx. So in terms of differential field

extension, we consider a differential field extension L = K(z)/K with z′ = z, z being

transcendental over K. So this is a particular example of Instructive Case (IC) of

Subsection 4.4. The elements of K are meromorphic over U = C and we choose the

reference point x0 = 0 ∈ C. Let now z(c : x) be the solution of (15) with initial

condition

(16) z(c : x0) = c,

where c is a parameter, so that

(17) ∂z(c : x)/∂x = z(c : x).

We can express concretely

(18) z(c : x) = c expx.

and hence we have a constraint

(19) c∂z(c : x)/∂c = z(c : x).

We notice here that we can obtain (19) without knowing the explicit form (18). In

fact, taking the partial derivative with respect to c of (17), we get

∂ (∂z(c : x)/∂c) /∂x = ∂z(c : x)/∂c,


314 H. UMEMURA

i.e., ∂z(c : x)/∂c also satisfies the differential equation (15). Since both z(c : x) and

∂z(c : x)/∂c satisfy (15),

∂(z(c : x) (∂z(c : x)/∂c)

−1)/∂x = 0

or

z(c : x) (∂z(c : x)/∂c)−1

is independent of x. So, there exists a function φ(c) of c such that

(20) z(c : x) = φ(c)∂z(c : x)/∂c.

Substituting x0 for x and using (16), we get φ(c) = c, hence (19) as promised. Now,

the new base field is a partial differential field

(K(c), {∂/∂x, ∂/∂c})

and we consider the partial differential field extension

K(c)(z(c : x))/K(c).

So, an element of the Galois group G-Galois (L/K) is a coordinate transformation

c 7→ ϕ(c)

of the space C of initial condition leaving the left hand side of

z(c : x) (∂z(c : x)/∂c)−1

= c

invariant, c being an element of the partial base field K(c). Namely,

ϕ′(c)−1ϕ(c) = c

or

cϕ′(c) = ϕ(c).

Consequently, ϕ(c) = λc, λ being a non-zero complex number. This means that the

coordinate transformation c 7→ ϕ(c) is c 7→ λc. Hence, it follows from (13) that the

Galois group

G-Galois (C(x, z)/C(x)) with z′ = z

is Gm = C∗. We have, moreover,

G-Galois (C(x, z)/C(x)) ' Aut(C(x, c, z(c : x))/C(x, c)) ' Aut(C(x, z)/C(x)),

where the middle term is the group of K(c)-automorphisms of the partial differential

field K(c, z(c : x)) with derivations ∂/∂x, ∂/∂c.



Example 2. — The argument of Example 1 allows us to show that for a Picard-

Vessiot extension L/K, Galois group G-Galois (L/K) coincides with the Galois group

G(L/K) of the Picard-Vessiot extension L/K. To be more precise, let K be a ordinary

differential field of meromorphic functions over a domain U of C with CK = C. Given

an n× n square matrix A ∈Mn (K), we consider a linear differential equation

(21) Y ′ = AY.

Replacing the domain U by a subdomain if necessary, we may assume that we can find

a solution Y (x) of the linear differential equation (21) meromorphic over the domain

U with detY 6= 0. Now, we choose a reference point x0 ∈ U and let Y (c : x) be a

solution containing the full parameters taking an appropriate initial conditions at the

reference point x0. Then, the argument in Example 1 shows that there exists an n×n

square matrix C = (cij) with cij ∈ C(c) for 1 ≤ i, j ≤ n and detC 6= 0 such that

(22) Y (c : x) = Y (x)C.

It follows from the equality (22) that the partial differential field

K(c) < Y (c : x) >

with derivations ∂/∂x, ∂/∂c generated by Y (c, x) over K(c) coincides with the field

K(c, Y (x)). In terms of differential field extension, we start from the ordinary dif-

ferential field extension K(Y (x))/K, which is a Picard-Vessiot extension, and pass to

the partial differential field extension

(23) K(c) < Y (c : x) >= K(c, Y (x))/K(c) with derivations∂

∂x,∂

∂c.

So, it follows form (13) that

G-Galois (K(Y (x))/K)

consists of the transformations c 7→ ϕ (c) of the space of initial conditions leaving all

the constraints invariant. Now, the argument of the previous Example shows that the

group G-Galois (K(Y (x)/K)) coincides with the automorphism group of the partial

differential field extension (23) and consequently to the Galois group of the ordinary

differential field extension K(Y (x))/K:

G-Galois (K(Y (x))/K) ' Aut(K(c) < Y (c, (x) > /K(c))

= Aut(K(c, Y (x))/K(c)) ' Aut(K(Y (x))/K).

Example 3. — Let us apply this idea to the first Painleve equation. Let us take as

the base field C(x) which we denote by K. Let us consider the first Painleve equation

(24) y′′ = 6y2 + x.

This means in terms of field extension that we consider a differential field extension

K(y, y′)/K such that y, y′ are transcendental over K and such that the derivatives

of y and y′ satisfy d(y) = y′ and d(y′) = 6y2 + x. So, this is a particular case of


316 H. UMEMURA

the Instructive Case (IC) of Subsection 4.4. We choose a reference point x0 ∈ C and

consider a solution y(c1, c2 : x) of the first Painleve equation (24) regular around x0

with initial conditions

(25) y(c1, c2 : x0) = c1, y′(c1, c2 : x0) = c2.

We show that the Jacobian

(26)J(y(c1, c2 : x), y′(c1, c2 : x))

J(c1, c2)= 1.

In fact, denoting the left hand side of (26) by F (c : x), we have

∂F (c : x)

∂x=

∂

∂x

∣∣∣∣∣∣∣∣

∂y(c : x)

∂c1

∂y(c : x)

∂c2∂y′(c : x)

∂c1

∂y′(c : x)

∂c2

∣∣∣∣∣∣∣∣

=

∣∣∣∣∣∣∣∣

∂y′(c : x)

∂c1

∂y′(c : x)

∂c2∂y′(c : x)

∂c1

∂y′(c : x)

∂c2

∣∣∣∣∣∣∣∣+

∣∣∣∣∣∣∣∣

∂y(c : x)

∂c1

∂y(c : x)

∂c2∂y′′(c : x)

∂c1

∂y′′(c : x)

∂c2

∣∣∣∣∣∣∣∣

=

∣∣∣∣∣∣∣∣

∂y(c : x)

∂c1

∂y(c : x)

∂c2

12y(c : x)∂y(c : x)

∂c112y(c : x)

∂y(c : x)

∂c2

∣∣∣∣∣∣∣∣

= 0.

So, F (c : x) is independent of x. It follows from (25) F (c, x) = F (c : x0) = 1 proving

(26). Hence, the Galois group G-Galois (L/K) is a Lie pseudo-subgroup of coordinate

transformations

(c1, c2) 7→ (ϕ1(c1, c2), ϕ2(c1, c2))

leaving the left hand side of (26) invariant. Namely

(27)J(y(ϕ1(c1, c2) : x), y′(ϕ2(c1, c2) : x))

J(c1, c2)=J(y(c1, c2 : x), y′(c1, c2 : x)

J(c1, c2).

Substituting x0 for x in (27), we get

(28)J(ϕ1, ϕ2)

J(c1, c2)= 1.

Conversely, if (28) is satisfied, since the both sides of (27) is independent of x, the

condition (27) is equivalent to condition (28). So G-Galois (K(y, y′)/K) is a Lie



pseudo-subgroup of the Lie pseudo-group consisting of all the transformations

(c1, c2) 7→ (ϕ1(c1, c2), ϕ2(c1, c2))

satisfying (28) or leaving the area invariant (cf. Question 5).

4.6. Technical refinement. — We started from an ordinary differential field ex-

tension L/K and constructed a partial differential field extension (14). We call

reader’s attention to the fact that the partial differential field

K(c, ∂|I|zi/∂cI)I∈Nn

depends on the reference point x = x0. Hence, we set

L|[x0] := K(c, ∂|I|zi/∂cI)I∈Nn

to show clearly its dependence on the reference point x0. We remark here two points.

First, Examples 1 and 2 show that in those cases the partial differential field L|[x0] is

independent of the reference point x0. Second, in Examples 1 and 2, the Galois group

G-Galois (L/K) is the automorphism group of the partial differential field extension

(14). In Example 3, however, besides the fact that it is not clear that the partial

differential field L|[x0] is independent of the reference point x0, the Galois group is

not the automorphism group of the partial differential field extension L|[x0]/K(c)

but it is a set of transformations leaving the area invariant. So, it is not a group

but a Lie pseudo-group. What about considering the automorphism group of the

partial differential field extension L|[x0]/K(c) in general? It is not a bad idea but it

means that since a differential field automorphism of L|[x0] is given by a birational

transformation c 7→ ϕ(c) of the space of initial conditions, we look for algebraic

transformations leaving the constraints invariant or satisfying a system of partial

differential equations such as (28). In the case of Example 3, we have sufficiently many

solutions of (28) in the birational transformation group of the plane, the Cremona

group of 2-variables. In general, however, we do not always have sufficiently many

algebraic solutions to the system of partial differential equations. In other words, the

automorphism group Aut(L|[x0]/K(c)) of the partial differential field extension might

be too small (cf. Remark 4.5 below). Hence, we can not limit ourselves to algebraic

solutions but we have to look for analytic solutions of the system of partial differential

equations of constraints. In the general case where the field of constants is not the

complex number field C, we can not speak of convergence so that we consider formal

solutions to the system of partial differential equations or we consider the continuous

differential automorphism group of a completion of L|[x0] with respect to a certain

topology.

Remark 4.5. — We examined the idea of considering a subgroup that is defined by a

system of partial differential equations, of the birational automorphism group of the

space of initial conditions. The birational automorphism group of an algebraic variety


318 H. UMEMURA

V defined over C, which is the C-automorphism group of the function field C(V ) is

small. In fact, let C be a non-singular projective curve defined over C of genus g. We

know

1. If g = 0, then Aut(C(C)/C) is isomorphic to PGL2(C).

2. If g = 1, then Aut(C(C)/C) is an algebraic group whose connected component

of the unit element 1 is isomorphic to the elliptic curve C.

3. If g ≥ 2, then Aut(C(C)/C) is a finite group of order d, where

d = 84(g − 1), 48(g − 1), 40(g − 1), 36(g − 1), · · · .

4.7. Infinitesimal automorphism group. — Now, we choose a point c0 in the

space of initial conditions or we choose a particular value

c0 = (c0 1, c0 2, · · · , c0 n) ∈ Cn

of c and we expand analytic functions of x and c around (c0, x0) into power series

with respect to local parameters

c0 := c− c0 = (c1 − c0 1, c2 − c0 2, · · · , cn − c0 n) ∈ Cn, x0 := x− x0.

In the sequel, when we consider the Taylor expansion of an analytic function at a

point, we say that we Taylor expand the function at the point. If there is no danger

of confusion, we omit suffix 0 and denote c0 and x0 respectively by c and x. In

particular the solution zi(c : x) of the ordinary differential equation (11) that is

regular at (c0, x0), is Taylor expanded into a power series of c, x. We have so far

realized the partial differential field L|[x0] as a partial differential subfield of the field

of of Laurent series:

L|[x0] → C[[c, x]][c−1, x−1].

So, we may write y(c, x) = y(c, x). We denote the image of the partial differen-

tial field L|[x0, c0] in C[[c, x]][c−1, x−1] by L|[c0, x0]. We consider the completion

L|[c0, x0]. of the partial differential field L|[c0, x0] with respect to the c-adic topol-

ogy. We can show that the completion L|[c0, x0] coincides with the closure, with

respect to the c-adic topology, of the field L|[x0 c0] in the field

C[[c, x]][c−1, x−1]

so that

L|[c0, x0] ⊂ C[[c, x]][c−1, x−1].

We would define the Galois group of the ordinary differential field extension L/K by

G-Galois (L/K)[c0, x0] := Aut(L|[c0, x0]/K(c)),



where Aut means the group of continuous K(c)-automorphisms of the partial differ-

ential field. We notice here that in the definition of the Galois group

G-Galois (L/K)[c0, x0],

we may replace the base field K(c) = K(c) by its completion K(c) = K[[c]][c−1] so

that we have

G-Galois (L/K)[c, x] := Aut(L|[c0, x0]/K(c)).

Let

Φ ∈ G-Galois (L/K)[c, x] := Aut(L|[c0, x0]/K(c)).

Identifying the solution zi(c : x) ∈ L|[x0] with its image z(c : x) in

L|[c0, x0] ⊂ C[[c, x]][c−1, x−1],

we may denote zi(c : x) by z(c : x). Since topologically and differentio-algebraically

the topological partial differential field L|[c0, x0] is generated over K(c) by the zi(c :

x)’s for 1 ≤ i ≤ n, the continuous automorphism Φ is determined by the images

Φ(zi(c : x)) that are elements of

L|[c0, x0] ⊂ C[[c, x]][c−1, x−1].

Since the zi(c : x)’s and Φ(zi(c : x))’s, which are elements of the field

C[[c, x]][c−1, x−1]

of Laurent series, are solutions of the ordinary differential equation (11), they would

differ by the initial conditions. There would exist a formal coordinate transformation

(29) c 7→ (ϕ1(c), ϕ2(c), · · · , ϕn(c))

such that

Φ(zi(c : x)) = zi(ϕ(c) : x) for 1 ≤ i ≤ n.

The transformation c 7→ ϕ(c) = (ϕ1(c), ϕ2(c), · · · , ϕn(c)) should satisfy a system of

partial differential equations so that

zi(c : x) 7→ zi(ϕ(c) : x) (1 ≤ i ≤ n)

determines a continuousK(c)-automorphism of the partial differential field L|[c0, x0].

This intuitive argument is almost correct but not rigorous and we need a technical

refinement. We regret that this procedure of justification makes the theory less ac-

cessible.

The above argument contains two problems. The first problem comes from the fact

that our guess that the transformation (29) is regular or equivalently it is given by a

set of formal power series is false. In fact they are formal Laurent series. So to have


320 H. UMEMURA

a correct picture, we must restrict ourselves to formal coordinate transformations.

Hence, we set

Aut0(L|[c0, x0]/K(c)) := {Φ ∈ Aut(L|[c0, x0]/K(c)) | Φ is induced

by a regular formal transformation (29)}.

To obtain more natural definition of Aut0(L|[c0, x0]/K(c)), we must replace the par-

tial differential field extension by a partial differential algebra extension. See Re-

mark 4.8.

To illustrate the second problem that we encounter, we consider the differential

equation (11) for n = 1. Suppose that in the differential equation (11) we have no

constraints. This happens in the most general case. Then the above argument gives

us if it were correct,

Aut0 (L|[c0, x0]/K(c)) = {ϕ ∈ C[[c]]|ϕ′(0) 6= 0}

The left hand side is a group by composition of maps but the right hand side is not

a group. In fact, let ϕ(c) and ψ(c) be two formal power series with coefficients in C,

then we can not always consider the composite ϕ(ψ(c)). If we calculate formally for

two formal power series

ϕ(c) =

∞∑

i=0

aici, ψ(c) =

∞∑

i=0

bici ∈ C[[c]]

the composite, we get

(30) ϕ(ψ(c)) = a0 + a1b0 + a2b20 + · · · (a1b1 + 2a2b0b1 + 3a3b

20b2 + · · · )c+ · · ·

that does not have any sense in the formal power series ring C[[c]] in c with coefficients

in C. The error of the argument comes from the fact that in general z(ϕ(c) : x) does

not belong to the field L|[c0, x0]. To remedy this, we consider only infinitesimal de-

formations of the identity automorphism of the partial differential algebra or in terms

of coordinate transformations we consider only those coordinate transformations that

are infinitesimally close to the identity.

For a commutative C-algebra A, we denote by N(A) the ideal of all nilpotent

elements of A. Let

ϕ(c) =

∞∑

i=0

aici, ψ(c) =

∞∑

i=0

bici ∈ A[[c]]

such that ϕ(c) and ψ(c) are congruent to the identity or to the power series c modulo

N(A). More concretely,

a0, a1 − 1, a2, · · · , b0, b1 − 1, b2, · · · ∈ N(A).

Then, the composition ϕ(ψ(c)) in (30) is a well-determined element of A[[c]].



4.8. Formal groups and group functors. — Let x1, x2, · · · , xn, y1, y2, · · · , yn

be variables over a commutative ring R. We denote formal power series rings

R[[x1, x2, · · · , xn]], R[[x1, x2, · · · , xn, y1, y2, · · · , yn]]

respectively by R[[x]], R[[x, y]]. A formal group of n-variables defined over R is an

n-tuple

F (x, y) = (F1(x, y), F2(x, y), · · · , Fn(x, y))

of formal power series Fi(x, y) ∈ R[[x, y]] of 2n-variables for 1 ≤ i ≤ n satisfying the

following conditions.

F (x, 0) = F (0, x) = 0.

F (F (x, y), z) = F (x, F (y, z)) for three sets of n-variables x, y, z.

For a formal group F (x, y) of n-variables, there exists a unique n-tuple

φ(x) = (φ(x1), φ(x2), · · · , φ(xn))

of formal power series φi(x) ∈ R[[x]] of n-variables for 1 ≤ i ≤ n such that φ(0) = 0

and such that

F (x, φ(x)) = F (φ(x), x) = 0.

Here are examples of formal groups of 1-variable.

F (x, y) = x+ y,

F (x, y) = x+ y + xy.

More generally, let G be a complex Lie group. Writing the group law G × G → G

locally at the unit element 1, we get a formal group. The above examples are particular

case of taking G = C, C∗.

Let F = F (x, y) be a formal group of m-variables and G = G(u, v) a formal group

of n-variables both defined over R. A morphism ϕ : F → G of formal groups is an

n-tuple

ϕ(x) = (ϕ1(x), ϕ2(x), · · · , ϕn(x))

of formal power series ϕi(x) ∈ R[[x]] of m-variables such that ϕ(0) = 0 and such that

ϕ(F (x, y)) = G(ϕ(x), ϕ(y)).

There is an elegant way of associating a group functor to a formal group. Let F

be a formal group of n-variables defined over R. We set

F(A) = N(A)n

and define a group structure on F(A) by putting

(a1, a2, · · · , an) · (b1, b2, · · · , bn) := F (a, b).

Since ai’s and bi’s are nilpotent elements of the commutative R-algebra A, F (a, b) is

a well determined element of F(A). This composition law defined on the set F(A) a

group structure. Indeed, the composition law is associative by the second condition in


322 H. UMEMURA

the definition of formal group, 0 is the unit element and the inverse a−1 of an element

a of F(A) is given by φ(a). We constructed a group functor F on the category (Alg/R)

of commutative R-algebras.

F : (Alg/R) → (Grp) := Category of groups.

We can prove the following

Proposition 4.6. — The functor associating to a formal group F the group functor F

is fully faithful. Namely, for formal groups F = F (x, y), G(u, v) defined over R, we

have

HomR(F,G) ' Hom(F, G),

where Hom in the right hand side is the set of morphisms of group functors.

4.9. Lie pseudo-group and Lie-Ritt functor. — Let

ϕ(x) = a0 + (1 + a1)x+ a2x2 + · · · , ψ(x) = b0 + (1 + b1)x+ b2x

2 + · · ·

be two formal power series in 1-variable x. Assuming that a1, a2, · · · , b1, b2, · · · are

variables, let us calculate the composite power series ϕ(ψ(x)) formally so that we get

ϕ(ψ(x)) = a0 + b0 + a1b0 + a2b20 · · ·+ (1 + a1 + b1 + a1b1 + 2b0(1 + b1)b2 + · · · )x+ · · · .

Setting formally the composite

ϕ(ψ(x)) := H0(a, b) + (1 +H1(a, b))x+H2(a, b)x2 + · · · ,

we have

H0(a, b) = a0 + b0 + a1b0 + a2b20 · · ·

H1(a, b) = a1 + b1 + a1b1 + 2b0(1 + b1)b2 + · · ·

· · · .

We can prove easily

Hi(a, b) ∈ Z[[a, b]] = Z[[a 0, a1, a2, · · · , b0, b1, b2, · · · ]].

with no constant term, i.e., H1(0, 0) = 0 for i = 0, 1, 2, · · · . Upon writing

H(a, b) = (H0(a, b), H1(a, b), · · · ),

we have

H(H(a, b), c) = H(a, H(b, c)).

c = (c0, c1, · · · ) being another of variables. So we can consider H = H(a, b) as a

formal group of infinite dimension defined over Z and a fortiori over C. We denote

this infinite dimensional formal group H by Γ1. The suffix 1 means that we deal with

transformations of 1-variable. We can associate a group functor

Γ1 : (Alg/Z) → (Grp)



to the formal group Γ1. It follows from the definition of the associated group functor

Γ1 (A) = {ϕ(x) ∈ A[[x]]|

ϕ(x) ≡ x modulo the ideal N(A) of nilpotent elements of A }.

Here, the group law is the composite of power series that are congruent to the identity

modulo the ideal N(A) of nilpotent elements. This is the group functor that we

introduced in Subsection 4.7. So far, we studied the 1-variable case. We can treat

the n-variable case similarly to get the infinite dimensional formal group Γn (a, b) of

n-variable transformations and the group functor Γn associated to it. We consider

not only the group functor Γn but also subgroup functors of Γn defined by a system of

partial differential equations. We call such group functors, or formal groups, Lie-Ritt

functors. So we replace a Lie pseudo-group by a Lie-Ritt functor.

Proposition 4.7. — We define a group functor

Inf -aut (L|[c0, x0]/K(c)) : (Alg/C) → (Grp)

in the following manner. For a commutative C-algebra A, we set

Inf -aut (L|[c0, x0]/K(c))(A) = {Φ ∈ Aut0 (L[c0, x0]⊗C[[c]]A[[c]]/K(c)⊗C[[c]]A[[c]]

|Continuous differential automorphism Φ is induced

by a formal power series ϕ ∈ A[[c]]

congruent to the identity automorphism modulo N(A) }.

In other words,

Inf -aut (L|[c0, x0]/K(c))(A) =

{ϕ ∈ Γn(A) | zi(c : x) 7→ zi(ϕ(c) : x) ( 1 ≤ i ≤ n) defines

a continuous differential algebra automorphism of

L[c0, x0]⊗C[[c]]A[[c]]/K(c)⊗C[[c]]A[[c]]}

Here, in the right hand side, the completion is taken with respect to the c-adic topology

and Aut denotes the group of continuous differential automophisms. Then, the group

functor Inf -aut (L[c0, x0]/K(c)) is a Lie-Ritt functor.

We denote the Lie-Ritt functor Inf -aut (L[c0, x0]/K(c)) by Infgal(L/K)[c0, x0]

and call it the infinitesimal Galois group of the differential field extension L/K with

respect to the point (c0, x0). We explained how we replace a Lie pseudo-group by

a formal group (of eventually infinite dimension) or by the Lie-Ritt functor that it

defines.


324 H. UMEMURA

Remarks 4.8(1) In the definition of Aut0(L|[c0, x0]) and the Lie-Ritt functor

Inf -aut (L[c0, x0]/K(c)),

we restricted ourselves to the infinitesimal regular formal transformations, which does

not seem natural. We can carry out this procedure more naturally if we use a differ-

ential subring, a model whose quotient field coincides with the given differential field.

Let us illustrate this for Instructive Case (IC) of Subsection 4.4. In this case, we take

R = C[x], S = R[z1, z2, · · · , zn] so that they are closed under the derivation and

their quotient field is respectively K and L. In other words, R and S are respectively

a model of K and L. In place of the partial differential field extension (14), we define

a partial differential subalgebra S of L by

S|[x0] := R[c, ∂|I|+lzi/∂xl∂cI ]l∈N, I∈Nn, 1≤i≤n

and we introduce a partial differential subalgebra R := R[c] of K[c] so that we have

a partial differential algebra extension S|[x0]/R. Then, we Taylor expand them with

respect to the local parameters c so that we have a morphism

S|[x0] → C[[c, x]].

We denote the image of S|[x0] by S[c0, x0] so that L|[c0, x0] is the quotient field of

S|[c0, x0] ⊂ C[[c, x]] ⊂ C[[c, x]][c−1, x−1].

We introduce the (c)-adic completion in S|[c0, x0] as in L|[c0, x0], the partial differ-

ential algebra extension S|[c0, x0]/R[c] defines a Lie-Ritt functor

Inf -aut (S[c0, x0]/R[c]).

Namely, for a commutative C-algebra A, we set

Inf -aut (S[c0, x0]/R[c])(A) := {Φ ∈ Aut(S[c0, x0]⊗C[[c]]A[[c]]/R[c]⊗C[[c]]A[[c]])

|Φ is congruent to the identity automorphism modulo N(A) }.

Then, we can show that that the infinitesimal automorphism Φ in the right hand side

is induced by a regular transformation so that we have

Inf -aut (L|[c0, x0]/K(c))(A) = Inf -aut (S[c0, x0]/R[c])(A)

for A ∈ Alg (C). Consequently, we have

Inf -aut (S[c0, x0]/ ˆK[c]) = Infgal(L/K)[c0, x0].

(2) We worked over the ordinary differential field extension (11)

L = K(z1, z2, · · · , zn)/K

under the assumption that the zi’s are algebraically independent. Modifying the

argument slightly, we can drop this assumption. So, we can attach a Lie-Ritt functor

Infgal(L/K)[c0, x0] to a general ordinary differential field extension (11).



(3) An important property of the Galois group Infgal is that it is big enough. We

can express this fact by saying that if an element of L which is a subset of L|[c0, x0] on

which the Galois group acts, is left invariant by the Galois group, then it is algebraic

over the composite field K(CL) of the base field K and the field CL of constants of

L. In fact, a principal homogeneous space with group Infgal(L/K)[c0, x0] is hidden.

See Remarks 4.16.

4.10. Galois group at the generic point. — For an ordinary differential field

extension L/K, we defined the Galois group Infgal(L/K)[c0, x0], which is a Lie-Ritt

functor over C, in grosso modo an algebraic group over C. The Lie-Ritt functor

Infgal(L/K)[c0, x0] depends on the chosen reference point (c0, x0) of the space of

initial conditions. We can expect that it is independent of the point (c0, x0). See

Questions 2 and 3.

Following the argument of Subsection 4.9 at the generic point of the space of initial

conditions, we get the Galois group Infgal(L/K) that is a Lie-Ritt functor over the

field L\. Here L\ is the underlying field structure of the differential field L. The

Galois group is canonically constructed but it is defined over L\.

In fact, in the definition of Infgal (L/K)[c0, x0], we chose a point x0 that is called

a C-valued point in the language of algebraic geometry and consider the Taylor ex-

pansion around the reference point x0 ∈ A1C = Spec C[x]. Let us carry it out at the

generic point. This is done by the universal Taylor expansion, which we are going to

explain. In Example of §5, we show the procedure concretely for the Instructive Case

(IC). Let (R, d) be an ordinary differential algebra over Q.

Definition 4.9. — Let A be a commutative Q-algebra. A Taylor morphism is a differ-

ential algebra morphism

(R, d) → (A[[X ]], d/dx).

When the differential ring (R, d) is fixed, among the Taylor morphisms

(R, d) → (A[[X ]], d/dx),

there exists the universal one. Namely we consider a map

i : R → R\[[X ]]

sending an element a ∈ R to its formal Taylor expansion.

(31) i(a) =

∞∑

n=0

1

n!dn(a)Xn.

We can check that i is a ring morphism and compatible with derivations d, d/dX . So

i is a Taylor morphism. The following Proposition is a consequence of the definition

of the morphism i.


326 H. UMEMURA

Proposition 4.10. — The Taylor morphism

i : R → R\[[X ]]

is universal among the Taylor morphisms. Namely for a commutative Q-algebra A,

we have a bijection

HomDiff-ring(R, A[[X ]]) ' HomRing(R\, A).

Proof. — In fact, there exists a natural correspondence between the elements of the

two sets. We denote the ring morphism A[[X ]] → A, g(X) 7→ g(0) of taking the

constant term by f0. For a Taylor morphism ϕ : R → A[[X ]], we associate the ring

morphism f0 ◦ ϕ : R\ → A, which gives a map from the left hand side to the right

hand side. Conversely, given a ring morphism ϕ : R\ → A, then ϕ naturally induces

a differential algebra morphism ϕX : R\[[X ]] → A[[X ]] and hence a Taylor morphism

ϕX ◦ i : R→ A[[X ]].

To understand the universal Taylor morphism, let us take the differential field

L = C(x, z) of Example 1 so that d(z) = z. Then, since dn(z) = z for n = 0, 1 2, · · · ,

it follows from definition (31) of the universal Taylor morphism, the image

i(z) =

∞∑

i=0

1

n!dn(z)Xn = z exp X.

So, the image of z is the general solution of the differential equation z′ = z containing

initial condition z at X = 0.

Now, let us come back to the general setting. Let L/K be an ordinary differential

field extension such that L is finitely generated over K as an abstract field. Let

z1, z2, · · · , zn ∈ L

be a set of generators of the abstract field L\ over K\. Let

i : L→ L\[[X ]]

be the universal Taylor morphism. The image of an elements of L is Taylor expanded

as in the convergent case. In particular the images i(zi)’s of the generators zi’s are

Taylor expanded. They contain parameters, initial conditions. We differentiate the

generators i(zi) with respect to the initial conditions to generate a partial differential

subalgebra. To this end let us take a transcendental basis

u1, u2, · · · , ul

of the abstract field extension L\/K\. The partial derivations

∂/∂ui ∈ Der (K(z)\/K\)

are uniquely extended to the derivations of L\/K\ which we denote by the same

symbol ∂/∂ui so that

∂/∂ui ∈ Der (L\/K\).



As it is well-known, the derivations ∂/∂ui’s form a basis of the L\-vector space

Der (L\/K\).

Definition 4.11. — We denote the partial differential field

(L\, ∂/∂u1, ∂/∂/∂u2, · · · , ∂/∂ul)

by L].

Now, we add in the power series ring L\[[X ]], the partial derivations ∂/∂ui’s oper-

ating on the coefficients of power series. In other words, we introduce L][[X ]] whose

derivations are the derivations ∂/∂ui’s and d/dX for 1 ≤ i ≤ d. We interpret the

universal Taylor morphism i as i : L→ L][[X ]].

Definition 4.12. — The partial differential algebra generated by i(L) and L] in L][[X ]]

with derivations ∂/∂ui’s and d/dX will be denoted by AL. We also introduce the

partial differential algebra AK generated by i(K) and L] in the partial differential

algebra L][[X ]].

Remark 4.13. — Since the partial derivations ∂/∂ui’s form a basis of the L\-vector

space Der (L\/K\) and since we added L] in the construction of AL and AK , the

partial differential algebra AL is independent of the choice of the transcendence basis

u1, u2 · · · , ul.

We would like to make the parameters or the initial conditions explicit so that

the generators i(zi)’s are expressed as power series with respect to the parameters as

in the local convergent case studied in Subsection 4.4. As in the ordinary case, we

have the universal Taylor morphism for a partial differential Q-algebra. Let j : L] →

L\[[U1, U2, · · · , Ul]] be the universal Taylor morphism for the partial differential field

L] so that we have

(32) a 7→∑

m=(m1,m2,··· ,ml)∈Nl

1

m1!m2! · · ·ml!

∂|m|a

∂um1

1 ∂um2

2 · · · ∂uml

l

Um1

1 , Um2

2 · · ·Uml

l

for an element a ∈ L]. So the morphism j is compatible with two sets of derivations

{∂/∂u1, ∂/∂u2, · · · , ∂/∂ul} and {∂/∂U1, ∂/∂U2, · · · , ∂/∂Ul}.

Thanks to the universal Taylor morphism j, we Taylor expand the coefficients of

power series to get a differential algebra morphism

L][[X ]] → L\[[U1, U2, · · · , Ul]][[X ]].

Hence, we get partial differential algebras

AK ⊂ AL ⊂ L\[[U,X ]]

with derivations

{∂/∂U1, ∂/∂U2, · · · , ∂/∂Ul, ∂/∂X}.

Now, we have arrived at our goal.


328 H. UMEMURA

Definition 4.14. — We denote the quotient field of AK AL respectively by K, L that

are partial differential subfields of L\[[U, X ]][U−1, X−1].

Lemma 4.15. — The partial differential subalgebras K ⊂ L of L\[[U, X ]][U−1, X−1]

are contained in a smaller differential subalgebra L\[[U, X ]][X−1].

Proof. — In fact, the differential subalgebras AK and AL are subalgebras of the field

L][[X ]][X−1] of Laurent series so that we can construct their quotient fields K′, L′ in

the field L][[X ]][X−1]. Then, the images of K′ and L′ under the Taylor expansion

morphism of coefficients

L][[X ]][X−1] → L\[[U, X ]][X−1] → L\[[U, X ]][U−1, X−1]

are respectively K and L.

Thanks to Lemma 4.15, we have an inclusion

K ⊂ L ⊂ L\[[U, X ]][X−1].

The completions K, L with respect to the (U)-adic topology coincide with their closure

in L\[[U, X ]][X−1] and consequently they define Lie-Ritt functor.

Using the partial differential field extension

K ⊂ L ⊂ L\[[U, X ]][U−1,W−1],

we can argue as we did in Subsection 4.9 with partial differential subfields

K|[c, x] ⊂ L|[c, x] ⊂ C[[c, x]][c−1, x−1]

to get the infinitesimal Galois group Infgal(L/K), which is a Lie-Ritt functor defined

over L\. for the given ordinary differential field extension L/K.

Remarks 4.16(1) For the Galois group Infgal(L/K), we can not expect Galois correspondence.

Indeed, whereas Infgal(L/K) is in general infinite dimensional, the field L is finitely

generated over K. We can show, however, that for a differential intermediate field

K ⊂M ⊂ L we have a canonical surjective morphism

Infgal(L/K) → Infgal(M/K),

which will play an important role for irreducibility questions (cf. Theorem (5.14),

[14]). For other properties of Infgal, see Theorem (5.16), [14].

(2) A principal homogeneous space of the group functor Infgal(L/K) is hidden (cf.

Theorem (5.10), [14]).



PART II

QUESTIONS

5. Fundamental questions on Galois theory

In Subsection 4.10, we defined Galois group Infgal(L/K) of which the construction

is canonical, depending only on the given differential field extension L/K. The Lie-

Ritt functor Infgal(L/K) is, however, defined over L\ which is evidently too big.

Question 1(*) . — Can we descend the Galois group Infgal(L/K) that is defined over

L\, to CK?

As we have no idea to answer this Question, we propose a remedy (cf. Questions

2 and 3). Let us assume that the field L is finitely generated over the field CK of

constants of K. Using the notation of Subsection 4.10, we can find subalgebras

R ⊂ K, S ⊂ L

closed under the derivations

d, ∂/∂ui ∈ Der(L\/K\) (1 ≤ i ≤ l)

such that R ⊂ S, the fields K and L are respectively the quotient field of R and S and

such that the algebras R\ and S\ are of finite type over CK . (Example below will help

the reader to understand what we do.) We can apply the argument of Definition 4.12

and what follows in Subsection 4.10, where we introduced the partial differential

algebras AL and AK , for the differential field extension L/K to the differential algebra

extension S/R so that we get partial differential algebras AR and AS that are partial

differential subalgebras of S\[[U, X ]] with derivations

{∂/∂U1, ∂/∂U2, · · · , ∂/∂Ul, ∂/∂X}.

Namely, the partial differential algebra generated by i(S) and S] in S][[X ]] with

derivations ∂/∂ui’s and d/dX will be denoted by AS . Similarly the partial differential

algebra AR is differentially generated by i(R) and S] in the partial differential algebra

S][[X ]]. Here S] denotes the partial differential algebra

(S, {∂/∂z1, ∂/∂z2, · · · , ∂/∂zl}).

and i : S → S][[X ]] is the universal Taylor morphism. Using the universal Taylor

morphism

j : S] → S\[[U1, U2, · · · , Ul]],

we Taylor expand the coefficients of power series to get a differential algebra morphism

S][[X ]] → S\[[U1, U2, · · · , Ud]][[X ]]


330 H. UMEMURA

We identify the differential subalgebras AR, AS ⊂ S][[X ]] with their images in

S\[[U, X ]] to get partial differential algebras

AR ⊂ AS ⊂ S\[[U, X ]] ⊂ L\[[U, X ]]

with derivations

{∂/∂U1, ∂/∂U2, · · · , ∂/∂Ul, ∂/∂X}.

We denote AR and AS respectively by R and S. So we obtained a partial differential

algebra extension

S/R

and Galois group

Infgal(S/R) = Aut(S/R)

that is a Lie-Ritt functor defined over the ring S\, as in Subsection 4.10. We have by

Lemma (4.5), [14]

Infgal(S/R) ⊗R\ L\ ' Infgal(L/K).

See also Remarks 4.8 (1). By Hilbert’s Nullstellensatz the set of CK -valued points is

dense in the algebraic variety SpecS\.

Question 2. — Does there exist a non-empty Zariski open subset W ⊂ SpecS such

that for every CK-valued point P of W , the reduction

Infgal(S/R) ⊗ S\CL(P )

is independent of P?

We could answer it affirmatively by the following argument. First, reduce to the

case CL = C by Lefschetz’ principle and use analytic continuation.

Example. — Let us understand concretely what happens by the Instructive Case

(IC) of Subsection 4.4. The reader will realize that the argument above at the generic

point is very close to that of Remarks 4.8 (1). In fact, since K = C(x) and since L =

K(z1, z2, · · · , zn), we can take R = C[x] and S = R[z1, z2, · · · , zn], the derivations

being d and ∂/∂zi ∈ Der (L\/K\) for 1 ≤ i ≤ l = n. Let us denote the image of the

zi’s under the composite morphism

S → S][[X ]] → S\[[U, X ]]

of the universal Taylor morphisms by Zi(U, X) for 1 ≤ i ≤ n. The zi’s being a

solution of the system of ordinary differential equations of condition 3 in Instructive

Case (IC), the power series Zi(U, X)’s in the Uj ’s and X with 1 ≤ j ≤ n, satisfy

the system of ordinary differential equation of condition 3, (IC) with respective to

the derivation ∂/∂X . In other words, the Zi(U, X)’s are a solution of the system of

ordinary differential equations of condition 3, (IC) containing the parameters Uj ’s.

Let us clarify their initial conditions. To this end, let

i : S → S\[[X ]]



be the universal Taylor morphism so that

(33) i(zi) = zi + Fi(z)X + · · · ∈ S\[[X ]]

by (31) and the system of ordinary differential equations of condition 3, (IC). We

notice also

(34) i(x) = x+X ∈ S][[X ]] ⊂ S\[[U, X ]]

by (32). The equality (34) shows that the universal Taylor expansion (33) is the

formal Taylor expansion of the analytic function zi in x at the generic point x or at

the reference point x0 = x. Let

j : S] → S\[[U ]]

be the universal Taylor morphism so that

(35) j(zi) = zi + Ui ∈ S\[[U ]]

by (32). Then, it follows from the definition of Zi, (33) and (35)

(36) Zi(U, X) = zi + Ui + j(Fi(z))X + · · · ∈ S\[[U, X ]].

So, the Zi(U, X)’s are a solution of the system of ordinary differential equations of

condition 3, (IC) with respect to the derivation ∂/∂X with initial conditions

Zi(U, 0) = zi + Ui ∈ S\[[U ]].

Therefore,

R = C[x][x +X ][z1 + U1, z2 + U2, · · · , zn + Un] ⊂ S\[[U, X ]],

S = C[x][x +X ][z1 + U1, z2 + U2, · · · , zn + Un][∂IZi(U, X)/∂U I]1≤i≤n, I∈Nn

⊂ S\[[U, X ]].

Points

c0 = (c01, c02, · · · , c0n) ∈ Cn, x0 ∈ C

being given, we have an C-algebra morphism

(37) S\ = C[x, z1, z2, · · · , zn]\ → C

that sends x to x0 ∈ C and zi to c0i ∈ C for 1 ≤ i ≤ n. The morphism (37) induces a

morphism

(38) S\[[U, X ]] → C[[U, X ]] ' C[[c, x]].

The latter isomorphism identifies Ui with ci = ci − c0i for 1 ≤ i ≤ n and X with x.

The image of Zi(U, X) by the morphism (38) is nothing but

zi(c, x) ∈ S|[c0, x0] ∈ C[[c, x]].

So

R⊗C[x, z1, z2,··· ,zn]\ C = R⊗S\ C ' R|[c0],

S ⊗C[x, z1, z2,··· ,zn]\ C = S ⊗S\ C ' S|[c0, x0].


332 H. UMEMURA

Consequently, we have

Infgal(R/S) ⊗S\ C ' Infgal(L/K)[c0, x0].

The equality

Infgal(R/S) ⊗S\ L\ ' Infgal(L/K)

holds too.

Question 3. — If CL is C, then is there a canonical isomorphism

Infgal(S/R) ⊗ S\C ' Infgal(L/K)[c0, x0]?

Above, we have affirmatively answered Question 3 for the Instructive Case, where

P = (c0, x0) ∈ SpecS\.

It is very natural to ask how Malgrange’s Galois theory [11] of foliations and ours

of differential field extensions are related.

First of all, comparison requires assumptions under which both theories work. So,

we propose to clarify how his idea and ours are related. Let L/K be an ordinary

differential field extension such that the field L\ is finitely generated over K\. Then

we have Infgal(L/K)[c0, x0] as we introduced in Subsection 4.9. On Malgrange’s

theory side, we need an analytic space and a foliation on it. In his theory, a particular

attention is paid to get not only a Lie algebra but also a global Lie pseudo-group. For

a comparison with our theory, however, we need only Lie algebra. Hence, the question

is local. To make explanation simple, let us limit ourselves to the Instructive Case

(IC) of Subsection 4.4. We use the notation of the previous Example after Question 2.

We have on the algebraic variety

SpecS = Spec C[x] ×C Spec C[z1, z2, · · · , zn] ' A1 × An

a foliation F defined by the system of ordinary differential equations of condition 3,

(IC). Let Y be a ringed space whose underlying topological space is the space C×Cn

with the usual topology and whose structure sheaf is the sheaf of rings of rational

functions regular on a given open set. Let X be the similar ringed space constructed

from Spec C[x]. So we have the projection morphism p : Y = A1 × An → X = A1

of ringed spaces. Let (x0, c0) be a point of Y = C × Cn. We choose a neighborhood

U of the point (x0, c0). So we have p|U : U → X . The foliation F on Y induces a

foliation F |U on U . We can speak of the Lie groupoid which we shall here denote

by Mgal (L/K), associated with the foliation F |U on the ringed space U . This is,

by definition, the smallest Lie groupoid defined over the ringed space U whose Lie

algebra contains the vector fields of leaves of the foliation F |U . Our question in a

concrete form is

Question 4. — Do we have an isomorphism of Lie algebras

Lie ((Mgal (L/K)[c0, x0]) ' Lie (Infgal(L/K)[c0, x0])?

Here Lie means the Lie algebra.



We indicate briefly the reason why we can expect this isomorphism. The projection

p : Y → X defines a transversal structure in the sense of Malgrange, 5.2, [11]. In the

definition of Infgal(S/R)[c0, x0], we take all the algebraic relations among the partial

derivatives

∂z|I|i (c, x)/∂cI for I ∈ Nn

over K[c]. So, they are the richest transversal structure defined over the ringed space

X (cf. loc. cit.).

It is easy to formulate this question in a general differential field extension L/K.

Question 5. — Behavior of Infgal under specialization.

It would be sufficient to express logically the following fact. If we specialize an

equation, we will have more constraints so that the Galois group would be smaller.

6. Questions related with an application of Galois theory

Question 6(*) . — Using the notation of Example 3, calculate the Galois group

Infgal(K(y, y′)/K) for a general solution y of the first Painleve equation. We can

ask the similar question for the other Painleve equations.

The Galois group Infgal(K(y, y′)/K) is conjectured since almost 100 years [4].

Namely it is the Lie pseudo-group of transformations on the plane leaving area in-

variant.

u = (u1, u2) 7→ (ϕ1(u), ϕ2(u))

with the Jacobian

J((ϕ1(u), ϕ2(u))/(u1, u2)) = 1.

We can formulate this in terms of Lie-Ritt functor without difficulty. ( cf. Example

3, Subsection 4.5.) It does not seem easy to prove the conjecture. Maybe, it requires

a new idea.

A paper of J. Drach written in 1914 is quite original. He asserts the equivalence of

the following two conditions for a function λ(t).

(i) λ(t) satisfies the sixth Painleve equation.

(ii) The dimension of the Galois group of the non-linear differential equation

dy

dt=y(y − 1)(t− λ)

t(t− 1)(y − λ)

is finite.

In the second condition, the Galois group of general algebraic differential equation is

involved. Namely the second condition depends on his infinite dimensional differential

Galois theory, which has been an object of discussion since he proposed it in his thesis

in 1898.

We proved that the first condition (i) implies the second (ii).


334 H. UMEMURA

Theorem 6.1. — Let λ(t) be a function of t satisfying the sixth Painleve equation. Let

K = C(t, λ(t), λ′(t)) which is a differential field with derivation d/dt. Let L = K(y)

be a differential field extension of K such that y is transcendental over K and such

that y satisfies

dy

dt=y(y − 1)(t− λ)

t(t− 1)(y − λ).

Then the Galois group Infgal(L/K) is at most of dimension 3.

Our proof depends on R. Fuchs’ system. Looking at our proof, B. Malgrange

showed it using the Jimbo-Miwa system [8](cf. [16]). .

Question 7(*) . — To give a solution of P6 such that

dim Infgal(L/K) ≤ 2.

More generally, to classify such solutions.

Maybe,

dim Infgal(L/K) ≤ 1

for Hitchin’s algebraic solutions λ of P6 related with the dihedral groups [7]. It is

natural to ask the following

Question 8. — The notation being as in the previous question, is the extension L/K

not embeddable in a strongly normal extension?

Noumi and Yamada [12] introduced a new Lax pair associated with so(8) that

defines P6. This system seems more natural than Fuchs’ or Jimbo-Miwa’s [8]. Indeed,

in the Noumi-Yamada system, all the Backlund transformations arise from gauge

transformations. So, we expect an affirmative answer to the following

Question 9. — Can we use the Noumi-Yamada system to prove Theorem 6.1 or to

answer Question 7?

We can expect it.

The Noumi-Yamada system describes a monodromy preserving deformation of a

linear system that has an irregular singular point. So, the Galois group of the linear

system remains invariant.

Question 10. — What is the deformation invariant Galois group G of the Noumi-

Yamada system? Is it small? In other words, is the Lie algebra LieG isomorphic to

a Lie subalgebra of sl2?

Question 11(*) . — To develop the idea of Drach or to clarify what Drach meant by

the converse of Theorem 6.1.



G. Casale [2] pointed out that we could not expect the converse. He also proposes

to prove that if Infgal(L/K) is finite dimensional, then λ has no movable singular

point.

7. Question on infinite dimensional Galois theory of difference equation

It is a mixed theory in the following sense. We start from a difference equation

and we get a Lie algebra or a formal group of infinite dimension in general. Or we

start from what is discrete and get a continuous invariant. The idea is simple. In

the definition of Infgal at the generic point in Subsection 4.10, we just replace the

universal Taylor morphism by the universal interpolation morphism, which we will

explain. Let us briefly sketch the idea.

A difference ring is a commutative ring R with operation of the additive group Z

on the ring R. Let us denote the automorphism R → R sending an element a ∈ R

to 1 · a ∈ R by ϕ. Since the automorphism ϕ : R → R determines the operation

of the additive group Z on R, we denote the difference ring R with operation of Z

by (R, ϕ). When there is no danger of confusion of the operation of Z, we denote

(R, ϕ) by R. When we emphasize that we consider the commutative ring R, we use

the notation R\. A morphism of difference rings is a morphism of rings compatible

with the operations of Z.

Definition 7.1. — For a commutative ring A, we set

F (Z, A) := {f : Z → A}

that is the ring of A-valued functions on Z.

The commutative ring F (Z, A) has a natural difference ring structure. Namely,

for a function f(x) ∈ F (Z, A), we define (ϕf)(x) = f(x+ 1) for x ∈ Z.

Definition 7.2. — Let A be a commutative ring. We call a difference morphism

(R, ϕ) → F (Z, A) an interpolation morphism.

Let (R, ϕ) be a difference ring. Then we have a canonical interpolation morphism

i : (R, ϕ) → F (Z, R\) sending an element a ∈ R to the function f(x) such that

f(x) = ϕx(a) for x ∈ Z. We call the canonical morphism the universal interpolation

morphism. A similar argument as for the universal Taylor morphism allows us to

show the following assertion.

Lemma 7.3. — The universal interpolation morphism is universal among the interpo-

lation morphisms. In other words, for a commutative algebra A, we have a natural

bijection

HomZ((R, ϕ), F (Z, A)) ' Homalg(R\, A).


336 H. UMEMURA

Let now L/K be a difference field extension such that the field L\ is finitely gen-

erated over K\. We attach to the extension L/K a Galois group InfgalD (L/K). Let

i : L → F (Z, L\) be the universal interpolation morphism. Let us denote by L] the

partial differential field (L\, {d1, d2, · · · , dl}), where the di’s (1 ≤ i ≤ l) form a basis

of the L\-vector space Der (L\/K\) of K\-derivations of L\. Now {d1, d2, · · · , dl}

operates on the values of functions, or we can consider F (Z, L]). Hence, we have now

i(L), L] ⊂ F (Z, L]). Here, we regard L] as the set of constant functions on Z. Let

us set DAL := the subalgebra of F (Z, L]) generated by i(L) and L] closed under the

set {d1, d2, · · · , dl} of derivations and Z-difference operator ϕ of F (Z, L]). Similarly,

DAK := the subalgebra of F (Z, L]) generated by i(K) and L] closed under the set

{d1, d2, · · · , dl} of derivations and Z-difference operator ϕ of F (Z, L]). We expand

elements of L] by the universal Taylor morphism j : L] → L\[[U ]]. So, we have

DAK ⊂ DAL ⊂ F (Z, L]) → F (Z, L\[[U ]])

We define the (U)-adic completions DAK and DAL respectively of DAK and DAL.

So finally we can define the Lie-Ritt functor InfgalD (L/K) on the category Alg (L\)

of L\-algebras.

Question 12(*) . — Can we calculate Infgal for the discrete sixth Painleve equation

qP6 of Jimbo-Sakai [9]?

The discrete Painleve equation qP6 has the conventional sixth Painleve equation

as a continuous limit. In general, what is discrete is more difficult. Yet we might have

a chance.

8. Arithmetic questions on Painleve equations

Today, about 100 years after the discovery, no one can doubt that the Painleve

equations define special functions. Here is a list of reasons why they deserve to be so

considered [15].

(i) They are irreducible to the classical functions.

(ii) They involve hypergeometric functions and their confluents.

(iii) They have combinatorial features. Particularly they are related with combina-

torics of Young diagrams in substantial way.

We call reader’s attention to the arithmetic nature of the Painleve equations that is

not yet widely recognized. We use the notation of Noumi and Yamada [12] for the

sixth Painleve equation. Let ε1, ε2, ε3, ε4 be variables over the ring C(x). We set

α0 = 1 − ε1 − ε2, α1 = ε1 − ε2, α2 = ε2 − ε3,

α3 = ε3 − ε4, α4 = 1 − ε3 + ε4.

The relation with the traditional notation is

α0 = κt, α1 = κ∞, α2 = κρ, α3 = κ1, α4 = κ0.



Let p, q be variables over the ring C(x)[ε1, ε2, ε3, ε4] so that

C(x)[ε1, ε2, ε3, ε4][p, q]

is a polynomial ring with coefficients in C(x)[ε1, ε2, ε3, ε4]. We extend the differential

algebra structure

(C(x)[ε1, ε2, ε3, ε4], d/dx)

to the overring C(x)[ε1, ε2, ε3, ε4][p, q] by

(39)

dq

dx=

∂H

∂p,

dp

dx= −

∂H

∂q,

where

H :=1

x(x − 1)

[p2q(q − 1)(q − x) − p ((α0 − 1)q(q − 1)

+α3q(q − x) + α4(q − 1)(q − x)) + α2(α1 + α2)(q − x)] .

We know that the Hamiltonian system (39) is equivalent to the sixth Painleve equa-

tion. So, we denote the Hamiltonian system (39) by P6(ε1, ε2, ε3, ε4). When the

variables ε1, ε2, ε3, ε4 take particular values e1, e2, e3, e4 respectively, we denote the

corresponding Hamiltonian system by P6(e1, e2, e3, e4).

Question 13(*) . — Is every rational or more generally algebraic solution (q(x), p(x))

of the sixth Painleve equation P6(e1, e2, e3, e4) defined over the field Q(e1, e2, e3, e4)?

A priori, there is no reason why they are rational over the field Q(e1, e2, e3, e4).

It seems, however, that no counter-example is known so far (cf. Boalch [1]). For a

logical formulation of Question (13), see the argument below.

A more plausible and weaker setting is as follows. Let e1, e2, e3, e4 be complex

numbers. Let q(x), p(x) be an algebraic solution of P6(e1, e2, e3, e4). Let R be

the Riemann surface of the algebraic functions q(x), p(x) so that since the sixth

Painleve equation has no movable singular point, we have a covering structure π :

R → P1C unramified over P\{0, 1, ∞}. The field of meromorphic functions C(R) has

a differential field structure coming from the covering map π. Let us formulate the

question rigorously. Ring theoretically we have a C(x)-differential algebra morphism

(40) f : C(x)(e1, e2, e3, e4)[p, q] = C(x)[p, q] → C(R)

sending q 7→ q(x), p 7→ p(x). By Belyi’s theorem, the Riemann surface R is defined

over Q. Namely, there exists a non-singular projective curve C over Q and a Q-

morphism ψ : C → P1Q

such that we have an isomorphism

R ' C ⊗Q C

over P1C.


338 H. UMEMURA

We ask

Question 14(*) . — Does the differential ring morphism f in (40) descend over

Q(e1, e2, e3, e4)?

Namely, does there exist a Q(e1, e2, e3, e4)-differential algebra morphism

f0 : Q(x)(e1, e2, e3, e4)[P, Q] → Q(e1, e2, e3, e4)(C)

such that

f = f0 ⊗Q(e1, e2, e3,e4) C?

References

[1] P. Boalch – Six results on Painleve VI, this volume.[2] G. Casale – On a paper of J. Drach, (2004), Email private communication.[3] J. Drach – Sur les equations du premier ordre et du premier degre, C. R. Acad. Sci.

Paris (1914), p. 926–929.[4] , Sur le groupe de rationalite des equations du second ordre de M. Painleve, Bull.

Sci. Math. 39 (1915), no. 2, p. 149–166, premiere partie.[5] B. Dubrovin & M. Mazzocco – Monodromy of certain Painleve-VI transcendents

and reflection groups, Invent. Math. 141 (2000), no. 1, p. 55–147.[6] M. R. Garnier – Sur des equations differentielles du troisieme ordre dont l’integrale

generale est uniforme et sur une classe d’equations nouvelles d’ordre superieur dont

l’integrale generale a ses points critiques fixes, Ann. Sci. Ecole Norm. Sup. 26 (1912),p. 1–126.

[7] N. J. Hitchin – Poncelet polygons and the Painleve equations, in Geometry and anal-ysis (Bombay, 1992), Tata Inst. Fund. Res., Bombay, 1995, p. 151–185.

[8] M. Jimbo & T. Miwa – Monodromy preserving deformation of linear ordinary differ-ential equations with rational coefficients. II, Phys. D 2 (1981), no. 3, p. 407–448.

[9] M. Jimbo & H. Sakai – A q-analog of the sixth Painleve equation, Lett. Math. Phys.38 (1996), no. 2, p. 145–154.

[10] E. R. Kolchin – Algebraic groups and algebraic dependence, Amer. J. Math. 90 (1968),p. 1151–1164.

[11] B. Malgrange – Le groupoıde de Galois d’un feuilletage, in Essays on geometry andrelated topics, Vol. 1, 2, Monogr. Enseign. Math., vol. 38, Enseignement Math., Geneva,2001, p. 465–501.

[12] M. Noumi & Y. Yamada – A new Lax pair for the sixth Painleve equation associatedwith bso(8), ArXiv: math-ph/0203029.

[13] H. Umemura – Galois theory of algebraic and differential equations, Nagoya Math. J.144 (1996), p. 1–58.

[14] , Differential Galois theory of infinite dimension, Nagoya Math. J. 144 (1996),p. 59–135.

[15] , Painleve equations in the past 100 years, in Selected papers on classical analysis(K. Nomizu, ed.), vol. 204, AMS Translations, 2001.

[16] , Monodromy preserving deformation and differential Galois group I, in Analysecomplexe, systemes dynamiques, sommabilite des series divergentes et theories galoisi-ennes, volume en l’honneur de Jean-Pierre Ramis (M. Loday-Richaud, ed.), Asterisque,vol. 296, Soc. Math. France, 2004, p. 253–269.



Remarks added on October, 30th 2006

This article was written in 2004. Here are recent developments in this branch.

(i) As for Question 4, the author proved that his theory is equivalent to Malgrange’s.The result will appear in a note in preparation.

(ii) With regard to Question 6, G. Casale succeeded in calculating the Galois groupof the first Painleve equation (G. Casale, Groupoıde de Galois de P1 et sonirreductibilite, to appear in Commentarii Mathematicii Helvetici). He also de-termined the Galois group of the Picard solution of the sixth Painleve equation.We know that in general, or to be more precise if it is not algebraic, the Pi-card solution is not classical. Yet its Galois group is finite dimensional afterCasale. We can recognize this phenomenon only through general differentialGalois theory, illustrating how useful the theory is.

(iii) The following paper of Casale replaces reference [2] of the original version of ourarticle. G. Casale, A note on Drach’s conjecture, in preparation, which we find aswell as his other papers in his home page (http://www.perso.univ-rennes1.fr/guy.casale/Article.htm).

H. Umemura, Graduate School of Mathematics, Nagoya University

E-mail : [email protected]


GALOIS THEORY AND PAINLEVE EQUATIONS´ by Hiroshi Umemura · GALOIS THEORY AND PAINLEVE EQUATIONS´...

Documents

Transcript of GALOIS THEORY AND PAINLEVE EQUATIONS´ by Hiroshi Umemura · GALOIS THEORY AND PAINLEVE EQUATIONS´...